Running head: PCK FOR PRESCHOOL MATHEMATICSmathcoaching.erikson.edu/.../04/McCray-PCKforPreschoolMathematics.pdfErikson Institute. PCK for Preschool ... PCK for Preschool Mathematics:

This manuscript is based on dissertation research completed under the direction of Dr. Jie-Qi Chen at Erikson Institute, and was funded by a Head Start Research Scholars Grant from the U.S. Department of Health and Human Services, Administration for Children and Families and a grant from the Chicago Public Schools.

Running head: PCK FOR PRESCHOOL MATHEMATICS

PCK for Preschool Mathematics: Teacher Knowledge and Math-Related Language

Contribute to Children’s Learning

Jennifer S. McCray

Erikson Institute

PCK for Preschool 2

Abstract

This study examines relationships between a new teacher interview designed to assess

teachers’ pedagogical content knowledge (PCK) for preschool mathematics, their math-

related language, and school-year gains in mathematics ability scores by children in their

classrooms. Twenty-six teachers and 141 children in Head Start programs in a large

Midwestern city participated. Analysis using hierarchical linear modeling (HLM) finds

significant relationships between scores on the new PCK Interview and gains in

children’s mathematics learning. In a new finding, frequency of teacher math-related

language is only significantly related to child outcomes when delivered outside the large

group setting; frequency of math-related language during “circle time” showed no

association. A final HLM model combining PCK Interview scores and teacher math-

related language outside the large group finds both contribute significantly and positively

to child gains, suggesting that teacher PCK is not entirely mediated through math-related

language as measured here.

PCK for Preschool 3

PCK for Preschool Mathematics: Teacher Knowledge and Math-Related Language

Contribute to Children’s Learning

In 1988, Deborah Ball’s dissertation demonstrated the importance of assessing

teachers’ pedagogical content knowledge (PCK) for teaching mathematics. Ball took

Shulman’s (1986) idea – that a particular kind of content knowledge is specially suited to

teaching – and used it as the basis for a mathematics interview. In it, teachers were

presented with elementary classroom scenarios in which a student misunderstood

something, and teachers were asked how they would respond. Teachers with good PCK

for teaching mathematics, Ball reasoned, would come up with representations of the

mathematics that clarify meaning and guide future thinking. Shockingly few of the

teachers Ball interviewed had even adequate responses to these scenarios; some teacher

responses were simply incorrect. In a further application of Ball’s interview, Ma (1999)

interviewed Chinese teachers. Though these teachers were less educated than their U.S.

counterparts, they offered rich, connected responses, demonstrating what Ma termed a

“profound understanding of fundamental mathematics” (p. xxiv). Ball’s interview

accomplished two important things for mathematics education: first, it provided clear

evidence that U.S. teachers’ PCK for elementary mathematics teaching was lacking; and

second, it suggested the types of content and pedagogy that might constitute good PCK,

providing clear direction for elementary teacher education.

Unfortunately, Ball’s interview and subsequent work related to PCK for

elementary mathematics offer little to those of us wishing to better understand preschool

mathematics and its effective teaching. There are profound differences in the knowledge

bases of elementary and preschool mathematics; in particular, elementary-level

PCK for Preschool 4

mathematics uses written notation to support mathematical thinking, enabling the

accomplishment of longer strings of procedures and acting as a sort of meta-cognitive

crutch. Preschool children are active mathematics learners, but not ready to understand,

let alone use, written arithmetic. Instead, three- and four-year-olds are primed to notice

and explore the quantitative relationships in the world around them, and to begin using

language and other forms of primary representation to crystallize information about

pattern, shape, space, size, and number. Because of this, the preschool pedagogical task

shifts from ensuring a connected, conceptual understanding of new mathematical

procedures such as “moving the decimal over one place,” to helping young children

recognize, name, and begin to experiment with the mathematics in their classroom

environment and their own actions upon it. These particularly-preschool instructional

tasks require a knowledge base distinct from the one Ball’s work has helped to identify.

Meanwhile, converging evidence from neuroscience, developmental psychology,

mathematics and science educators, and early childhood professionals has helped to

create an emerging sense that preschool mathematics education in the U.S. is in need of

substantial improvement. It has become clear that early education is a more important

teaching opportunity than was previously understood. Recent research clearly

demonstrates the profound impact of early environmental influences on long-term brain

development (Markezich, 1996), suggesting the potential of early education to create

long-term benefits. Additionally, policymakers have become aware that more U.S.

children are attending early care and education programs before formal schooling than

ever before (U.S. Department of Education, 2000) and that good early education can have

beneficial long-term learning outcomes and economic benefits (Barnett, 2008; Horton &

PCK for Preschool 5

Bowman, 2002). Intervention that begins early has reported effects that extend into later

years, apparently creating opportunity for greater educational and other gains in the long

run (Bowman et al., 2001; Clements, Sarama, & DiBiase, 2004). Further, early

intervention specifically focused on mathematics has been shown to have broad positive

effects on student learning (Fuson, Smith, & Lo Cicero, 1997). These findings suggest

that preschool education in the U.S. offers an important national opportunity, and that

mathematics education merits special attention.

Other evidence suggests that the troubling tendency of American students to “lag

behind” their counterparts in other industrialized nations on tests of mathematics

(National Research Council, 1989, 1990; Schoenfeld, 1992) begins very early. Cross-

national differences have been found in not only the early elementary school grades (e.g.,

Frase, 1997; Stevenson, 1987; Stevenson & Stigler, 1992) but also at the preschool level,

where children from other developed and developing countries outperform their

American counterparts on such beginning mathematics concepts as number words and

early addition (e.g., Geary, Bow-Thomas, Fan, & Siegler, 1993; Ginsburg, Choi, Lopez,

Netley, & Chi, 1997; Starkey, Klein, Chang, Dong, Pang, & Zhou, 1999). Growing

evidence suggests that higher quality teaching of mathematics in the earliest years of a

child’s education offers the best hope for improving mathematics achievement in the U.S.

(Bowman et al., 2001; Clements, Sarama, & DiBiase, 2004).

While the education literature indicates that teacher quality is the most effective

predictor of student achievement (Darling-Hammond, 2000; Just for the Kids and The

Southeast Center for Teaching Quality, 2002; National Commission on Teaching &

America’s Future, 1996; Rice, 2003), the quality of teaching at the preschool level is

PCK for Preschool 6

extremely variable (Copple, 2004), and mathematical teaching is no exception. A joint

position statement on preschool math by the National Association for the Education of

Young Children and the National Council of Teachers of Mathematics (NAEYC, 2005) is

specific about the problem. The two organizations note the general lack of good teacher

preparation in mathematics and point out that this under-preparation contributes to poor

math-related attitudes among many early childhood teachers who lack confidence in

mathematics anyway. While lack of confidence does not, in itself, prevent a teacher from

teaching math, it can feed an unfortunate tendency to avoid math in the classroom. A

recent contribution to preschool mathematics teacher training goes so far as to devote a

section to this issue, noting “Math Anxiety – You Can Handle It” (Smith, 2001, p.2), and

when surveyed (Carpenter, Fennema, Peterson, & Carey, 1988), both pre- and in-service

teachers in early childhood classrooms expressed great reluctance to teach mathematics,

making comments like “I don’t do math.” Describing early childhood educators, Copley

notes “to them, mathematics is a difficult subject to teach and one area that they often

ignore except for counting and simple arithmetic” (2004, p. 402). There is good reason to

assume that the teaching of preschool mathematics could be improved.

PCK for Preschool Mathematics – Construction of the Teacher Interview

In an attempt to better illuminate what preschool mathematics is and make clear

the importance of teaching it knowledgeably and well, the author designed a new teacher

interview to assess PCK for preschool mathematics. Literature on elementary

mathematics education, preschool pedagogy, and mathematics content for very young

children were consulted to create a tool that might allow teachers to demonstrate the

knowledge that undergirds excellent preschool mathematics teaching. Where there was a

PCK for Preschool 7

lack of relevant preschool educational literature, ideas from cognitive developmental

theory and recent research on quantitative development in early childhood were

incorporated. The concept of pedagogical content knowledge (PCK), however, provided

the guiding framework for the interview’s content and construction.

According to Lee Shulman, who first identified pedagogical content knowledge

(PCK) during his 1985 presidential address to the American Educational Research

Association (Shulman, 1986), PCK is “a knowledge of subject matter for teaching which

consists of an understanding of how to represent specific subject matter topics and issues

appropriate to the diverse abilities and interest of learners” (Shulman & Grosman, 1988,

p. 9). For example, PCK emphasizes knowledge of which content ideas are more central

to the subject and how they connect to one another (subject matter understanding),

appropriate examples for illustrating those concepts (teaching techniques), and awareness

of how those concepts develop in the thinking of novices with differing levels of

experience (knowledge of student development). In other words, knowledge of content,

teaching practice, and student development are effectively combined where good PCK

exists. See Figure 1 for a Venn diagram illustrating how these three bodies of knowledge

overlap to construct PCK.

The new Preschool Mathematics PCK interview assessed here (see Appendix A)

borrows Deborah Ball’s 1988 innovation of situating interview questions within teaching

scenarios and asking teachers to report what specific actions they would take in each

situation (see also Study of Instructional Improvement, 2002). Using teaching scenarios

rather than items solely about content knowledge contextualizes questions so they are

more like the kinds of dilemmas teachers actually encounter in the classroom. This

PCK for Preschool 8

approach provides a mechanism for tapping content knowledge, pedagogy, and

understanding of student learners simultaneously, acting as a better approximation of

PCK than a purely content-related question could ever supply.

Figure 1. Pedagogical Content Knowledge (PCK)

Subject Matter Understanding

Knowledge of Students’

Development

Teaching Techniques

Pedagogical Content

Knowledge

Importantly, Ball’s questions are also designed to assess teachers’ ability to help

students of elementary mathematics maintain and enhance connections between

procedures and concepts in their mathematical thinking. The idea that there are two

distinct types of mathematical knowledge – procedural knowledge, which encompasses

the forms, rules, and processes that make it possible to accomplish mathematical tasks,

and conceptual knowledge, which embodies richly connected ideas about the

mathematical relationships between things and actions – is an old one. U.S. mathematics

education has a long history of emphasizing first one of these types of knowledge, and

then the other, with little effective result (see, e.g., McLellan & Dewey, 1895; Thorndike,

1922; Wheeler, 1939). It was not until the early 1980s that Resnick and Ford suggested

PCK for Preschool 9

that rather than trying to determine their relative importance to math education, the

relationships between concepts and procedures ought to be emphasized (1981). In

alignment with this thinking, Ball’s interview pointedly asks elementary mathematics

teachers to come up with real-world representations that can illustrate the usefulness and

meaning of specific mathematics procedures.

For example, in one interview scenario, pre-service teachers were presented with

the division statement 1 ¾ / ½ (“one and three quarters divided by one half”). Ball

asked the pre-service teachers to “develop a representation – a story, a model, a picture, a

real-world situation” of this division statement (Ball, 1988, p.16). An appropriate model

of this statement would demonstrate that it will answer the question “how many halves

are there in one and three quarters?” The effective teacher, Ball reasoned, would be able

to describe a situation in which such a question is meaningful. As an example of a useful

representation, Ball suggests “A recipe calls for ½ cup of butter. How many batches can

one make if one has 1 ¾ cups butter? Answer: 3 ½ batches” (1988, p. 16). Interview

questions such as this made clear which teachers had more of the kind of knowledge that

would enable them to keep mathematical procedures meaningfully connected to the ideas

they are meant to represent.

The concept-procedure literature in mathematics education is ambivalent about its

own usefulness for preschool mathematics. Interestingly, it defines preschool

“procedures” and elementary (and above) “procedures” extremely differently, though the

ramifications of these differences are largely unexamined (see, e.g., Hiebert & Lefevre,

1986). In the elementary-based literature procedures are often algorithmic processes

enacted upon symbols such as “borrowing from the 10s column,” while in the preschool

PCK for Preschool 10

version of this analysis procedures are generally physical actions taken and enacted upon

concrete objects. For example, in the concept-procedure literature, a toddler uses the

“procedure” of placing a spoon in each teacup to construct a working concept of one-to-

one correspondence (see Sinclair & Sinclair, 1986). From this perspective, very young

children’s mathematics concepts are completely dependent upon their procedures;

mathematical knowledge is being created through action and has not yet been separated

from the world of things. Sinclair and Sinclair comment “the young child cannot do

without actual experience when logico-mathematical knowledge is in its beginnings”

(1986, p. 63). As in elementary mathematics, connections between concepts and

procedures in mathematical thinking are central; however, instead of being at risk of

becoming separated from one another, perhaps they are at risk of not being firmly

established in the first place.

The relatively sparse current literature on early mathematics education seems to

support this perspective. Clements notes “although young children possess rich

experiential knowledge, they do not have equal opportunities to bring this knowledge to

an explicit level of awareness” and suggests the importance of differentiating between

“the intuitive, implicit, conceptual foundation for later mathematics” and the subsequent

elaboration that produces something more like conceptual math knowledge (2004, p. 11).

Copley seems to agree, suggesting that conceptual learning requires something beyond

actions taken upon concrete objects alone. She notes “Early childhood educators say that

children learn by doing. The statement is true, but it represents only part of the picture.

In reality a child learns by doing, talking, reflecting, discussing, observing, investigating,

listening, and reasoning” (2000, p. 29).


If preschoolers can benefit from assistance in establishing and consolidating

initial connections between their budding mathematical ideas and more generalized

concepts, this provides an initial portrait of what good preschool mathematics teaching

must be. Following the thinking of children as they interact with materials, recognizing

the mathematical potential in their activities, and knowing how to comment on and

extend their mathematics-related thinking must all be central. To provide opportunities

for teachers to demonstrate these abilities, the new interview presents teachers with two

classroom-based free play scenarios – one in the dramatic play area and one in the block

corner. First, teachers read the scenarios to themselves, then the researcher reads them

aloud, and finally, the researcher asks the teacher to do various things, including:

identify specific math-related topics the children’s play addresses; suggest a comment

they might make to help the children think about/become more aware of the mathematics

in their play; and propose a question they could ask that might encourage children to

experiment with the mathematics in their play and extend their thinking. In this way, the

interview is meant to elucidate preschool teachers’ abilities to help young children

construct and consolidate their initial mathematical concepts.

Each of the mathematics content strands proposed by the National Council of

Teachers of Mathematics (NCTM) in its Principles and Standards for School

Mathematics (2000) is represented in each scenario either through the use of specific

materials, the comments children make during the scenario, or the problems children

encounter and actions they take. For example, Scenario 1 presents teachers with the

following text:

Brittany and Jacob are playing in the dramatic play area and want to put their five babies to bed. There are no doll beds, so they make “cribs” out


of three shoeboxes. Jacob says “but there aren’t enough cribs.” Brittany responds, “these babies are younger” picking out the three babies with no hair and setting them near the shoeboxes. She picks up the two babies with thick hair, says “these babies don’t need to nap anymore,” and sets them aside. Jacob says “OK, but this baby needs the most room” and puts the biggest bald baby in the biggest shoebox. Brittany watches him and then puts the medium-sized bald baby in the medium-sized shoebox and the smallest bald baby in the smallest shoebox. Jacob says “now go to sleep, babies.” (Appendix A)

In this scenario, the small, medium, large relationships of the babies and shoeboxes

represent a repeating pattern, while the idea of using a shoebox for a crib is an example of

three dimensional geometric thinking. Recognizing that there “aren’t enough cribs”

requires one-to-one correspondence, which is foundational to number sense. When

Brittany solves this dilemma by putting aside the two babies who have hair, she classifies

and sorts a single set (babies) into two sets (bald babies and babies with hair).

Measurement skills are activated when Jacob determines which baby needs the most

room, and which shoebox offers it. In the interview, teachers who can see more

mathematics in such play and generate effective comments that point it out and

encourage its elaboration score more points.

A small group of experts in quantitative development and preschool pedagogy

assessed the face validity of the interview, and contributed to a list of possible teacher

responses. The list was used to construct response rubrics for each interview question

which allow interviewers to check off key items as they are mentioned by teachers. Early

piloting among three teachers indicated more and different responses than anticipated.

Response rubrics were revised and resubmitted to the experts for comment; their

comments were incorporated. Further piloting among six preschool teachers suggested


the interview was sensitive enough to produce sufficient variability with a range of scores

from 12 to 34 and a standard deviation of 7 (scores between 0 and 86 being possible).

Teacher Math-Related Language

The study also explores the relationship between this construction of preschool

mathematics PCK and teachers’ practices by examining teacher math-related language.

Recent work (Ehrlich, 2007; Klibanoff, Levine, Huttenlocher, Vasilyeva, and Hedges,

2006) has found a positive association between frequency of preschool teachers’ use of

mathematics-related talk and gains in mathematics their students make during the school

year. Assuming this finding would be replicated, coding systems from Klibanoff, et al.,

(2006) were applied to the classroom speech of the teachers interviewed so the predictive

power of the interview could be compared to that of math-related language (see

Appendix B).

The conceptualization of PCK above, however, suggested further differentiation

of the relationship of math-related language to child outcomes might be possible.

Teacher math-related language can appear at many times during a preschool day and in

many different settings. It is often a part of the whole classroom meeting, or circle time,

when preschoolers may count the number of children present or review the calendar and

count off the date. It can also, however, be present when teachers work with a small

group of children to complete a project, as in “each ladybug gets two eyes,” or when

teachers are supporting children’s play, as in “four of those cups are little, and two are

big.” Since the representation of good mathematics pedagogy developed above

suggested children develop mathematical concepts while operating directly on the world

around them, it was theorized that circle time might not be the best opportunity for


mathematical language to aid in this endeavor. Teacher math-related language offered

while children are actively engaged in their own pursuits, however, could be differently

received and processed because aligned with children’s actions or mathematical

“procedures.” To assess whether pedagogical setting had any bearing on the relationship

between math-related language and child learning, teacher speech data was gathered in

both large group settings (circle time) and in free play and small group settings (non-

circle time).

Research questions were:

1) Is there a significant association between teachers’ PCK interview scores

and gains children in their classrooms make?

2) Is there a significant association between teachers’ math-related speech

and gains children in their classrooms make (replicating the findings of

others)? Is the strength of this association essentially the same for math-

related speech during circle time and outside the circle time setting?

3) Is the relationship between PCK and child gains mediated by teacher

math-related language, and if so, to what extent?

Methodology

Participants

Twenty-six Head Start teachers from a large urban area and 113 children from

their classrooms with assessment scores at two time points participated. To be

considered for the study, teachers had to work in classrooms in which English was not a

second language for a majority of students and be “Head” teachers in their classrooms.

Teachers were selected at random from among this group and recruited for participation;


14 CPS teachers agreed to participate. The author randomly selected non-CPS Head Start

agencies to contact from a list provided by City of Chicago staff. Twelve non-CPS Head

Start teachers agreed to participate. Participating teachers were given a $100 stipend and

their classrooms received a set of picture books after teacher interviews were complete.

Classroom size ranged from 13 to 21 children, with a mean of 18 and a standard

deviation of 2.5. While Head Start admits children between the ages of 3-0 and 4-11 at

the start of the school year, on average, classrooms in the sample had five three-year-olds

in September of 2006. Overall, 70% of the students in participating classrooms were

African-American, 23% were Latino, and the remaining students were of other ethnic

origin, including Brazilian, Caucasian, Chinese, Czech, German, Hmong, Polish, and

Vietnamese. Taking overall classroom composition into account, more than half the

classrooms served an all African-American population, several served a mostly-Latino

population, and the remainder served a very diverse population. On average, 23% of

students in these classrooms spoke English as a second language. The 26 classrooms

were clustered within 19 sites. Fourteen sites held only one participating classroom, four

sites had two participating classrooms, and one site held four.

In order to participate, children had to be between 3-4 and 5-0 years old, and be

fluent in English, according to the teacher. When possible, six such children were

randomly selected to participate from among those whose parents had consented in each

classroom. One hundred forty-one children were assessed at T1 in the 26 classrooms that

participated for the length of the study, or an average of 5.4 children per classroom.

These children averaged 4 years and 4 months of age at first testing, and 76 of them, or

53.9%, were female. One hundred ten children, or 78.1%, were African-American, 26


were of Latino origin, and 5 were identified as Asian/Pacific Islander. Of these 141

children, only 113 had mathematics assessment scores at two time points (others having

moved away or dropped out of their class). Only these children are included in the

analysis, resulting in an average of 4.3 children per classroom.

The missing subjects were generally demographically similar to the entire sample.

They averaged 4 years and 2 months of age, 90% came from English-only homes, and

their ethnic make-up was strikingly consistent with the sample as a whole, with 80% of

the group African-American, 16% Latino, and 4% Asian. They were, however, more

likely to be female: 73.9% of these T1-only subjects were female, compared with 53.9%

of the sample as a whole. Because of this, the remaining sample includes a higher

percentage of male children (53%) than the original sample (46.1%).

Teacher Language Samples

Participating teachers’ speech was tape-recorded and coded on one randomly

selected day during the period from the beginning of January 2007 until end of April

2007. One hour of teacher speech was recorded and coded, always before noon, and

included both “circle-time” (teacher-led large group) and the period immediately

following; recording/coding continued until 60 minutes were captured. Activity after

circle time consisted of combinations of free play, in which teachers supported children’s

activities in areas of their own choosing, and small-group activities, in which teachers

met with a small group of children to guide them through a teacher-led set of activities.

To facilitate recording, each teacher wore a wireless microphone, which broadcast a

signal to be coded and recorded. Two researchers were present during recording to code

teacher math-related language responses; a timer was utilized to prompt researchers to


take turns coding in 10 minute intervals. Teachers were unaware of the mathematics-

focus of the study when the language sample was recorded and coded.

The 60 minutes of coded teacher language began with the teacher’s first address

to the whole group of gathered children at the beginning of circle time. Coding

categories from Klibanoff, et al. (2006) were used to code teacher math-related language

(see Appendix B for category definitions). This coding system heavily emphasizes

references to number and quantity – geometric and spatial math-related language is not

explicitly represented. Because live coding of math-related language using this coding

system had not been previously attempted, all language samples were recoded by the

graduate student researcher using audiotapes, yielding an inter-rater reliability score of

96.4%; disagreements were settled by discussion.

Since language data was collected on only one day for each classroom, an

examination of total math-related language instances was conducted to screen for

outliers. That is, teachers who chose to emphasize mathematics on the one day of

language recording far beyond what they would generally do needed to be accounted for,

since the purpose of the language data was to find a representative sample of math-related

language for each classroom. Using the commonly accepted formula of

< Q1 – (1.5 x IQR) or > Q3 + (1.5 x IQR)

in which Q1 = 1st quartile score, Q3 = 3rd quartile score, and IQR = Inter-quartile range

(3rd quartile score – 1st quartile score), four teachers’ frequency of math-related language

scores were determined to be outliers; these four teachers and the data retrieved from

within their classrooms (e.g., child mathematics gains) were removed from the data set

for all further analysis.


Upon review of the audiotapes of these outlier teachers, it was discovered that one

teacher spent the entire free play time going over number flash cards with children,

another played a 25-minute game of number bingo, a third used almost 80% of the 60

minutes to play “find the number” with a small group of children (a game in which she

said, for example, “where is the two?” and children picked out the correct numeral from

a display), while the fourth spent her entire non-circle time working on ordinal

relationships with a small group of children (saying, for example, “three, four, five...what

comes next?”). While these activities may not represent unexpectedly advanced

mathematics teaching, the dramatic impact they have on these teachers’ language

instances on the single day of recording and coding make the scores unlikely to be

representative of their usual practice. Further, since these outlier classrooms represent

the most math-related language delivered outside the circle time setting, their removal

makes the assessment of non-circle time language as a predictor of child outcomes a

more stringent one.

Test of Early Mathematics Ability

To assess gains Head Start children made in mathematics during the study period,

the Test of Early Mathematics Ability, 3rd edition, (TEMA-3) (Ginsburg & Baroody,

2003) was administered twice – once in the fall and once in the spring. The TEMA-3 is a

standardized, norm-referenced instrument designed to assess both informal (non-school

taught) and formal mathematics achievements among 3 to 9 year old children. Basals

and ceilings are utilized in the TEMA-3, and testing begins with an item corresponding to

the child’s age and ends when the child makes five consecutive errors. Children are

asked to count the objects on a card, select the card with more objects and point to it,


show a specific number of fingers, perform basic level mental addition and subtraction,

and write numbers, among other things. TEMA-3 has two parallel forms with established

test-retest reliability; children were randomly assigned Form A or Form B at T1 and

received the alternate form at T2.

In 2002, the TEMA-2 (Ginsburg & Baroody, 1990) was reviewed as part of a

workshop on “school readiness” sponsored by the U.S. Department of Health and Human

Services, and was found to be “based on current theory and research…grounded in

normative data…(and) a good predictor of mathematics achievement” (U.S. Department

of Health and Human Services, 2002, p. 23). The normative sample for this assessment

was representative of the national population in terms of sex, race (black, white, and

other), region of the country, residence in an urban or rural community, and parent

occupation (white-collar, blue-collar, service, farm, or other). Cronbach’s Alpha for

TEMA-3 math ability scores at age four is reported to be .93 for Form A and .95 for

Form B, well within the most desirable range for reliability. Test-retest reliability is

similarly high, with a delayed alternate-form coefficient of .93 (Ginsburg & Baroody,

2003).

Testing took place in a quiet setting near the child’s classroom, often at a small

table in the hallway just outside the room. Sometimes a separate room, down the hall

from the classroom, was the option provided by the school. Assessments were conducted

by graduate students in child development. Most TEMAs took approximately 15 minutes

to administer, though assessment times ranged from about 2 minutes to almost 47

minutes, depending upon the child’s ability and willingness to participate.

Results


Is There a Significant Association between Teachers’ PCK Interview Scores and Gains

Children in Their Classrooms Make?

Interview reliability. Because the Preschool Mathematics PCK Interview is a new

measure, analyses were conducted to assess its internal reliability before examining its

relationship to teacher practices. As a first measure of reliability, correlations were run

between total Scenario scores. Scores for Scenarios 1 and 2, each meant to reflect PCK

across mathematics content areas, correlated strongly (r = .616, p < .001). Utilizing sub-

scenario scores, Cronbach’s alpha for the items comprising these two Scenarios was a

strong .76, representing an acceptable level of inter-item reliability (according to

Nunnaly, 1978, the standard is .7.) Interviews were coded live, and fifteen (50 %) were

coded simultaneously by two researchers. Inter-rater reliability was 92.8%;

disagreements were settled by discussion and used an interview audiotape. Teachers

scores ranged from 6 to 32 points, with an average score of 21.6 and a standard deviation

of 5.8.

Children’s gains in mathematics. At Time 1, the 91 children in non-outlier

classrooms scored between 65 and 132 on the TEMA-3, with a mean score of 85.11 and a

standard deviation of 12.43. This mean score is nearly 1 standard deviation below the

population average of 100, indicating the sample children were doing noticeably worse

on mathematics than their same-age peers nationally. At Time 2, scores had not changed

that much overall. The children scored between 62 and 124, with a mean of 86.11 and a

standard deviation of 14.74. In fact, six children showed no change in TEMA-3 score

between time points, and 71.4% had scores that changed less than 10 standard points in

either direction. Further, 53.8% of the sample showed either no change in score, or a


negative change in score. Recall that the TEMA-3 is standardized for age so that simply

getting the same test items correct from one session to the next is not enough to maintain

the same standardized score. The lack of change reflected in some of these scores may

represent an actual increase in knowledge and/or more items correct on the assessment

while also indicating no movement relative to national averages for same-age peers.

Regardless, this sample of Head Start children is not keeping up nationally, and by this

measure, does not show an overall trend to improve its relative standing.

Relationship between interview scores and child gains. A 3-level Hierarchical

Linear Model (HLM) was used to examine the relationship between PCK Interview

scores and gains in child scores from T1 to T2 (see Table 1). HLM is generally

considered appropriate whenever data has a nested structure, as it does in this study, since

child outcomes are clustered by teacher and teachers are clustered by program sites.

HLM offers several advantages over simple regression analysis in this case. First, while

regression techniques allow examination of relationships between teacher PCK and child

outcomes within classrooms, they neglect to account for variance related to program site.

That is, because we know at which site each teacher is located, if we do not conduct

analyses that account for site-level variance, we are not using all the available

information. At least some of the variance in child outcomes is likely to be due to

program site, rather than teacher; HLM allows us to attribute that variance accurately

while assessing relationships between teacher PCK and child outcomes, yielding greater

precision. Moreover, HLM allows the study to specifically examine the effects of

program site and how those effects interact with teacher effects. That is, we can


determine whether and how site characteristics might affect relationships between teacher

language and child outcomes.

In effect, HLM creates one linear regression model at each level of analysis and

then combines these equations into a single model. In this study, child variables (gains in

TEMA-3 scores) are at Level 1, teacher variables (math-related language frequencies and

interview scores) are at Level 2, and while no site variables are used as predictors, the use

of a third level allows an examination of the effects of program sites. In this way, the

modeled “levels” mimic the hierarchical clustering of the actual children, classrooms, and

sites. In Level-1 of the example below, the gains of any particular child (TEMDIF) are

modeled as the mean gains of all children in that classroom (π0) plus some unique effect

associated with the particular child whose gains are being modeled (e). The Level-2

model is meant to predict the mean gains of all the children in a given classroom (π0) as

the sum of the mean gains of all children at that particular program site (β 00 ) plus a

unique effect associated with that particular classroom (r0). Level-3 models the mean

gains of all children at a given program site (β 00 ) as the grand mean of all children’s

gains (γ000 ) plus a unique effect due to program site (u00).

Level-1 Model

TEMDIF = π0 + e

Level-2 Model

π0 = β00 + r0

Level-3 Model

β 00 = γ000 + u00

These 3 levels yield a combined model of:


TEMDIF = γ000 + u00+ r0+ e

This explanatory model of gains on the TEMA-3 is sometimes called a fully

unconditional model (see Raudenbush & Bryk, 2002, p. 24) because no predictors (or

conditions) are specified. It is generally helpful to run such a model as a preliminary step

because it provides baseline information about variability in the outcome variable (in this

case, TEMDIF, or children’s TEMA-3 gains) at each level of analysis. Results of the

fully unconditional model for TEMDIF indicate that 19.6% of the variance between child

gains can be attributed to differences at the classroom level. Since this study is focused

on the impact of teachers’ knowledge and practice upon child outcomes, this is the

portion of the variance in child gains that is of interest. The fully unconditional model

also indicates that this variance is significant, X2 = 25.86 with 6 df (p <.000), but variance

between sites is not, X2 = 24.31 with 15 df (p = 0.06, ns). In other words, while

classroom makes a significant contribution to variation among children’s TEMA-3 gains,

program site does not. Teachers (or something at the classroom level) appear to be

having more of an impact upon children’s mathematics learning than program sites.

To evaluate whether teacher PCK for preschool mathematics as measured by the

new teacher interview can explain some of this classroom-level variance, interview

scores were entered in the model as predictors at Level-2 (ITONETWO), resulting in the

following model construction:

Level-1 Model

TEMDIF = π0 + e

Level-2 Model

π0 = β00 + β01(ITONETWO) + r0


Level-3 Model

β 00 = γ000 + u00

β 01 = γ010

When interview scores are entered into the model as a predictor, they significantly

and positively predict gains in child outcomes; that is, the higher the teacher’s PCK score,

the greater the gains children in her classroom are likely to have made from T1 to T2 (see

Table 1).

Table 1

Results of 3-Level HLM Models to Predict Child Gains on TEMA-3 (TEMDIF) Preschool Mathematics PCK Interview as a Predictor Fixed Effect Coefficient

Standard Error T-ratio df

p-value

For Intercept 1, π0 Intercept 2, β00 -9.521734 4.032275 -2.361 15 0.032 ITONETWO, β01 0.496883 0.180434 2.754 20 0.013

Is There a Significant Association between Teachers’ Math-related Speech and Gains

Children in Their Classrooms Make? Is the Strength of this Association Essentially the

Same for Math-related Speech Both During Circle Time and Outside the Circle Time

Setting?

Math-related language. Total frequency of math-related language for the 22

teachers ranged from 4 to 74 instances (M = 30.86, SD = 20.65). Circle time math-

related language ranged from 0 to 50 instances (M = 18.95, SD = 16.01), and non-circle

time math-related language ranged from 0 to 34 instances (M = 11.91, SD = 9.65). Circle


and non-circle time frequencies did not correlate with one another (r = .248, p = .133,

ns); that is, frequency of math-related language used during circle time did not

significantly predict non-circle time frequency, and vice versa. Circle time ranged in

length from almost 6 minutes to nearly 42 minutes (M = 20.95, SD = 11.87). As might

be expected, teachers with longer circle-times used more instances of math-related

language during that time (r = .673, p < .000). This relationship did not hold true for

non-circle time, however (r = -.144, p = .261, ns); that is, teachers who spent more of the

60 minutes outside of circle time did not also tend to have more non-circle time math-

related language instances. Frequency of math-related language during a particular

pedagogical setting, then, cannot be interpreted as a measure of time on task.

Additionally, the lack of correlation between amount of time spent and frequency of

math-related language outside of circle time suggests this measure is more sensitive than

the circle time measure to individual teacher variation.

Relationship between teacher language and child outcomes. Two 3-level

hierarchical linear models (HLM) were conducted to examine the predictive power of

teacher math-related language (see Table 2). In the first, to examine whether total

frequency of math-related language during the 60 minute period accounts for variance at

the classroom level, total math-related language (TOT) is entered into the otherwise fully

unconditional model as a predictor. Surprisingly, total frequency of math-related

language shows no significant relationship to child gains. As an alternative, the second

model enters both circle time language (CIRCTOT) and non-circle time language

(NONCIRC) simultaneously as Level-2 predictors. Accordingly, the model’s results

report the amount of variance explained by each of these variables after the other is


accounted for. This construction of the model is guided by the finding that circle time

language and non-circle time language do not correlate with one another, and are

therefore more likely to represent distinct contributions to variance among child

outcomes.

Level-1 Model

TEMDIF = π0 + e

Level-2 Model

π0 = β00 + β01(CIRCTOT) + β02(NONCIRC) + r0

Level-3 Model

β 00 = γ000 + u00

β 01 = γ010

β 02 = γ020

The results indicate that non-circle time math-related language has a significant

positive relationship to gains in child outcomes, but circle-time language does not

contribute significantly (see Table 2). To give a rough sense of the size of this effect, an

increase of one math-related language instance during the 60 minutes coded was

associated with a 0.26 increase in growth in TEMA-3 standard score (p < .003). Even

after the variance attributed to this model is accounted for, child gains continue to differ

significantly by teacher (X2 = 22.01, p < .000), indicating there is still unexplained

classroom-level variance.


Table 2

Results of 3-Level HLM Models to Predict Child Gains on TEMA-3 (TEMDIF) Teacher Math-Related Language as a Predictor Fixed Effect Coefficient


p-value

Total Math-Related Language Instances (TOT) in 60 Minutes as a Predictor

For Intercept 1, π0 Intercept 2, β00 -1.460862 2.387392 -0.612 15 0.549 TOT, β01, 0.083139 0.062128 1.338 20 0.196

Circle Time (CIRCTOT) and Non-Circle Time (NONCIRC) Math-Related

Language in 60 Minutes as Predictors

For Intercept 1, π0 Intercept 2, β00 -1.939136 2.130691 -0.910 15 0.377 CIRCTOT, β01, -0.024915 0.070202 -0.355 19 0.726 NONCIRC, β02 0.309822 0.109686 2.825 19 0.011

Is the Relationship between PCK and Child Outcomes Mediated by Teacher Math-related

Language, and if so, to What Extent?

One final model was run to determine whether PCK and teacher math-related

language have any distinct predictive power relative to child gains, or if instead, one of

these contributors loses significance when variance due to the other is accounted for. In

this model, PCK interview scores (ITONETWO), circle time language (CIRCTOT), and

non-circle time language (NONCIRC) are all entered simultaneously as Level-2, or

classroom-level, predictors. The resulting Combined PCK and Language Model (see


Table 3) indicates that both PCK and non-circle time language make independent

contributions to the prediction of child outcomes, while circle time language remains

non-predictive.

Table 3 Results of 3-Level HLM Model to Predict Child Gains on TEMA-3 (TEMDIF) Combined PCK and Language Model Fixed Effect Coefficient


p-value

For Intercept 1, π0 Intercept 2, β00 -10.729137 3.382006 -3.172 15 0.007 CIRCTOT, β01, -0.102650 0.065273 -1.573 18 0.133 NONCIRC, β02 0.261790 0.097992 2.672 18 0.016 ITONETWO, β03 0.502984 0.171150 2.939 18 0.009

Discussion

The results indicate that higher teacher scores on the PCK for Preschool

Mathematics Interview are significantly and positively associated with greater child gains

on the TEMA-3 between T1 and T2. While the total frequency of math-related language

during a 60 minute period showed no significant relationship to child gains, a subset of

this variable, math-related language delivered outside the circle time setting, proved

positively related. Math-related language delivered during circle time showed no

significant association with child gains. Finally, the PCK interview score and teacher

math-related language outside the circle time setting appear to act at least semi-

independently as predictors of children’s gains, since each remained significantly and


positively associated with greater increases on the TEMA-3 even when variance

attributed to the other was simultaneously accounted for.

Math-Related Language

While findings reported here generally support the idea that more teacher math-

related language can positively influence children’s mathematical learning – a conclusion

that echoes those of Klibanoff et al. (2006) and Ehrlich (2007) – they also offer an

additional insight. Specifically, it appears that pedagogical setting can interact with

math-related language, impacting its effectiveness. In this study, only math-related

language delivered outside of the large group, circle time setting shows a significant,

positive relationship to child learning. More math-related language during circle time

shows no association with children’s improvement. Further, combining the two language

measures dilutes the effect: Total frequency of math-related teacher language is not

significantly related to child outcomes.

This effect of pedagogical setting is a new finding, not reported in previous

literature on teacher math-related language. Klibanoff et al. (2006) collected language

samples identical in form to those of this study – 60 minutes beginning with circle time –

but made no distinction between pedagogical settings in their analysis. In their work,

unlike this, teachers’ total frequency of math-related language was a significant predictor

of child learning. Ehrlich’s (2007) study provides an even stronger contrast, since her

language samples consisted exclusively of circle time, and again showed a significant and

positive relationship. In Klibanoff et al., however, the effect size found is very small,

since an increase of 25 instances of teacher math-related language during the 60 minute

observation period would result in only .21 standard deviations of achievement gain


(2006). While Ehrlich finds a more sizeable effect, in which an increase of one language

instance per minute results in 2.29 standardized points of gain in growth of the TEMA-3

score, this finding is only barely statistically significant (p < .056). Findings here for

non-circle time language, in contrast, are of both reasonable size and high significance

since an increase of only one math-related language instance during this period was

associated with a 0.26 increase in growth in TEMA-3 standard score (p < .002). To

provide a better comparison with Klibanoff, et al., an increase of 25 instances of teacher

math-related language outside the circle time setting would result in .70 standard

deviations of achievement gain, or more than three times as much. This comparison of

effect sizes and significance levels suggests it is at least possible that the inclusion of

circle time in language measures used by Klibanoff et al., (2006) and Ehrlich (2007)

diluted the observable effects of language upon child outcomes.

It can also be argued that Klibanoff et al.’s (2006) and Ehrlich’s (2007) classroom

populations may not be a good match to those of this study in terms of SES and program

quality. While Klibanoff et al. specifically examined SES and found no relationship to

either teacher practices or child outcomes, they note their sample was weighted toward a

high-SES population (2006). Of their 26 classrooms, only four were low SES, eight were

middle-income, and 14 were high-SES. Hence, it is possible their sample was not

representative enough to illuminate such differences, if they existed. Ehrlich’s work

directly assesses the effects of SES on the teacher language-child outcomes relationship

by comparing results for Head Start and tuition-based preschool classrooms; she also

finds no significant relationship between SES and the practice-outcomes relationship

(2007). However, the Head Start classrooms that comprise her lower income group are


located in Chicago Public Schools. This means they necessarily employ bachelor’s level

staff and offer a fairly rich and well-stocked classroom environment because of school

funding and other institutional supports.

The classroom sample in this study includes not only Chicago Public School Head

Start classrooms but also community-based Head Start classrooms, which are extremely

variable in terms of teacher education and resources. At many of the community-based

sites in this work, manipulatives were missing or broken, posters and visual displays were

of poor quality, children had no or limited access to outdoor play spaces, and classroom

space was limited. Often, it appeared that children spent much of their day supervised by

assistant teachers who rarely held even associate’s degrees; attention from the qualified

teacher (whose language sample was recorded) was more limited. Community-based

sites also frequently housed more than one Head Start classroom within a single large

room, so the general noise and activity level was much higher than in most Chicago

Public School classrooms. It can be legitimately argued that previous work on math-

related teacher language did not look at this specific population of preschool classrooms.

Less resource-rich classrooms, such as this study’s overall sample includes, may make

the relationship between math-related language and mathematics learning more

vulnerable to interactions with pedagogical setting. Further work is required to evaluate

this theory.

Post-hoc review of audiotapes indicates almost all the non-circle time language in

this sample is delivered as an accompaniment to children’s free play activity. Teachers

comment on what children say, as in “Yes, you’re right… you have three babies,”

involve themselves directly in children’s play, as in “is there one more plate for me?” and


extend children’s thinking, as in “what would happen if we used two blocks instead of

three?” In these types of situations, children are in the midst of play – manipulating

materials, imagining scenarios, and constructing representations of their own. Teacher

language serves to reflect, introduce, and extend the mathematical thinking embedded in

these activities. Regardless of whether it is better overall pedagogy, children’s self-

directed engagement, teachers’ individualized attention, or more likely, all these elements

in concert that is at work in these non-circle time interactions, findings here provide

persuasive evidence that teachers’ math-related interactions with children during their

free play have a powerful impact on their mathematical learning.

PCK Interview

Findings related to the interview suggest it functioned successfully as a measure

of knowledge for teaching preschool mathematics. As expected, teachers with higher

PCK Interview scores were more likely to teach children who made greater gains on the

TEMA-3. While the interview may not be the only or best way to assess PCK for

Preschool Mathematics, it appears to capture knowledge for teaching with a real

relationship to learning among children. The ability of preschool teachers to see,

comment on, and extend the mathematics they see in children’s play is clearly associated

with gains in their students’ mathematical understanding. At least to some extent, the

interview has successfully transferred the important work of elementary mathematics

education researchers such as Hill, Rowan, and Ball (2005) to a preschool setting by

providing a measure of mathematical content knowledge that is strongly associated with

children’s learning.


The final HLM model demonstrates that simultaneously accounting for variance

in child gains associated with teacher math-related language (while it marginally reduces

the size of the effect), actually increases the significance level of the relationship between

PCK Interview Scores and child outcomes. This is especially surprising since the

interview scenarios depict the very types of pedagogical settings in which non-circle time

math-related language was collected. One might expect that a different measure of

mathematics PCK that focused on, for instance, teachers’ knowledge for planning large

group activities, would be more likely to show a relationship to child outcomes that is not

mediated by non-circle time math-related language. Because the interview is entirely

focused on teacher-child interactions that occur outside circle time, it seems probable that

the addition of non-circle time math-related language to the HLM model would subsume

the contribution of PCK. Instead, we find each of these measures – one of teacher

practice, and one of teacher knowledge – are significant predictors in their own right. In

fact, one point on the interview contributes almost twice as much to variance in child

gains as one instance of teacher math-related language (see Table 3). Frequency of math-

related language cannot substitute for teacher knowledge, nor is it the case that frequency

of math-related language is the only, or even major, way that PCK for preschool

mathematics is expressed. In other words, while increasing the frequency of teacher talk

about mathematics may be one way to augment children’s mathematical understanding,

knowing more about the specific mathematics they are discovering and how to help them

see it appears equally and independently important.

While greater PCK predicted greater TEMA-3 gains, this does not speak to the

overall quality of teachers’ responses. When presented with scenario 1, in which children


place babies in shoebox “cribs” of different sizes, teachers were quick to see the

mathematics of measurement, but rarely saw that the cribs could elicit thinking about

shape and spatial relationships. When asked to comment on the block play depicted in

scenario 2, teachers tended to note the relevance of shape, but very seldom mentioned the

importance of classifying types of blocks or using number to count blocks in an effort to

build walls of the same length. The PCK of the study’s teachers appears to be supporting

the mathematical learning that does occur in these classrooms; that does not mean,

however, that it could not be radically improved. Particularly when we remember how

far behind their same-age peers nationally children in these classrooms are, these teacher

responses to the interview might be viewed as indications of how preschool educators

could improve their mathematical PCK and thereby children’s mathematical learning.

Limitations and Future Research

This study has several important limitations. Because the findings reported here

are associational, not causal, it is possible that some unmeasured factor, such as teacher

attitude toward mathematics, is the real engine behind children’s learning. Controlled

randomized design and replication of the results for both math-related language and the

PCK Interview are needed to confirm causal relationships. Further, the PCK Interview is

limited in scope, since it only directly addresses teacher knowledge used in free play

settings. While the teacher interview scores may in fact be good indications of teacher

knowledge for other kinds of preschool mathematics teaching, such as during circle time,

we have no way of knowing this. Without some effective measure of PCK for leading

large group explorations of preschool mathematics, it is unclear what contribution such

knowledge might make. Finally, the Klibanoff, et al. coding system (2006) is constrained


to teacher language that relates to number: measurement, spatial relations, geometry, and

pattern are generally ignored. While this coding system is impressive in its robust

relationship to gains in children’s learning, its limitations in scope may impede its ability

to capture much of teacher language that is of consequence. It may be that a broader

coding system could show a greater mediating relationship between teacher math-related

language and PCK for preschool mathematics.

Replication of the finding that pedagogical setting impacts the relationship

between teacher math-related language and child outcomes is needed. If, in fact,

pedagogical setting grows in influence on math-related teaching as SES or classroom

quality declines, this is an important finding with many practical implications for

intervention. There is some fairly robust work suggesting that circle time is not the most

effective mechanism for addressing content in preschool classrooms generally and that

children tend to learn more when engaged in free play (Montie, Claxton, & Lockhart

2007). However, given the universality of circle time in U.S. preschool education, more

in-depth examination of large group teaching techniques and their effects across socio-

economic groups may be in order. Further, circle time may serve important social and

behavior management functions that allow other learning to take place; assessing for

these differential types of effects could provide very useful information for early

childhood educators.

Conclusion

The concept-procedure analysis of mathematical knowledge helps illuminate the

central aim of effective mathematics education: namely, imparting a system of

mathematical thinking in which procedures and concepts are well-connected. This


educational goal works as well for preschool children as for elementary school children.

The difference is that mathematical procedures at these two levels of development take

very different forms. While elementary mathematics teachers must strive to impart

notation and algorithms (procedures) in a manner that emphasizes their meaningful

relationship to mathematical ideas about the world (concepts), preschool teachers must

work to provide language that captures key relational abstractions (concepts) just as

children encounter them through their actions upon objects (procedures). For

preschoolers, effective mathematics teaching must acknowledge the central importance of

their initial construction of mathematical concepts from experiences in the world and so

focus on naming and making that mathematics explicit.

Clements recommends mathematical teaching that “involves reinventing,

redescribing, reorganizing, quantifying, structuring, abstracting, generalizing, and

refining that which is first understood on an intuitive and informal level” (2004, p. 12).

Like Rogoff’s cognitive “apprenticeships” (1990), this type of teaching can be thought of

as a socialization process, in which young children, led by expert adults, learn to see and

think about the world in terms of quantitative relationships for the first time. In turn, the

meaningfully explicit mathematics of the preschooler should serve as a solid foundation

for the application of numeracy and notation that constitute elementary school arithmetic.

The findings reported here support the view that making mathematical thinking more

explicit and meaningful by connecting children’s actions to mathematical language can

have a lasting impact on their learning during a single school year, providing a better

foundation for later mathematical learning.


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Appendix A

PCK for Preschool Mathematics Teacher Interview


PCK for Preschool 45 PCK for Preschool 45









Appendix B

Math-Related Language Coding System from Klibanoff et al., 2006

1) Counting – encompasses both reciting counting words and counting objects in

sets.

2) Cardinality – involves stating (or asking for) the number of things in a set without

counting them. If cardinality is used to reinforce counting, it is coded as a

separate instance, e.g., “One, two, three. There are 3 books” would be coded as

two instances, one of counting and one of cardinality.

3) Equivalence – encompasses statements describing a quantitative match, either of

number or of amount, between two or more entities. These include: (a) one-to-

one mapping, e.g., each child gets one cracker; (b) one-to-many mapping, e.g.,

each group has four children; (c) stating that two amounts or sets are the same.

4) Nonequivalence – encompasses statements of two or more entities being unequal,

whether referring to (a) unspecified amounts, e.g., “Who has the most?” (b) one

amount specified and the other(s) unspecified, e.g., “Oh no, you have more than

twelve teeth” (c) all relevant amounts specified, e.g., “Seven people said yes, ten

people said no. Did more people say yes or say no?”

5) Number symbols - coded if utterances include instances in which a teacher labels

a written number symbol, or asks a child to identify, write, or find a number

symbol, e.g., “3” in a stack of cards with printed numbers.

6) Conventional nominatives – numbers used as labels for things or dates.


7) Ordering – instances of referral to a sequence with explicit reference to more than

one entity or set. Note that reciting a list of number words in order would not be

coded as ordering but rather as counting.

8) Calculation – includes cases in which a teacher performs a calculation or asks a

child to solve a calculation problem.

9) Place-holding – encompasses any input that refers to place value: ones, tens,

hundreds, etc., including, but not limited to, the decomposition of (at least) two-

digit numbers.

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